A376092 10^n-th powerful number.

1, 49, 3136, 253472, 23002083, 2200079025, 215523459072, 21348015504200, 2125390162618116, 212104218976916644, 21190268970925690248, 2118092209873957381248, 211765852717674823741924, 21174572668805230623003225, 2117363857447354911021280900

Offset: 0

Author

Chai Wah Wu, Sep 09 2024

Keywords

• nonn

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..16

Crossrefs

Cf. A001694, A118896.

Programs

Python

from math import isqrt
from sympy import mobius, integer_nthroot
def A376092(n):
    def squarefreepi(n):
        return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    def bisection(f, kmin=0, kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    m = 10**n
    def f(x):
        c, l = m+x, 0
        j = isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        c -= squarefreepi(integer_nthroot(x,3)[0])-l
        return c
    return bisection(f,m,m)

Formula

a(n) = A001694(10^n).

Limit_{n->oo} a(n)/10^(2n) = (zeta(3)/zeta(3/2))^2 = 0.21172829478335...